An active cloaking strategy for the scalar Helmholtz equation in three dimensions is developed by placing active sources at the vertices of Platonic solids. In each case, a “silent zone” is created interior to the Platonic solid and only the incident field remains in a defined region exterior to this zone. This distribution of sources ensures that implementation of the cloaking strategy is efficient: once the multipole source amplitudes at a single source location are determined, the other amplitudes are calculated by multiplying the multipole source vector by a rotation matrix. The technique is relevant to any scalar wave field.

## 1. Introduction

Over the last two decades, excitement has been generated around the idea of rendering objects invisible to incident excitation (Miller, 2006; Norris, 2008; Schurig , 2006), via passive (Cai , 2007; Silveirinha , 2007; Zhang , 2011) or active (Miller, 2006; Vasquez , 2009a; Zheng , 2010) cloaking. The latter employs sources to suppress fields and has been of interest since Lueg's patent in 1936 (Guicking, 1990). Significant work on anti-sound/anti-vibration has been carried out since then (Cheer, 2016; Fuller , 1996; Guicking, 2007; Nelson and Elliott, 1991). Specific choices for the active sources ensure quiet zones and illusions (Ma , 2013; Zheng , 2010) and control arrangements can be optimised to reduce acoustic scattering in a stationary fluid (Eggler , 2019a; House , 2020; Lin , 2021; O'Neill , 2015) and in a convected flow field (Eggler , 2019b).

Recent interest has centred on object-independent active cloaking methods, e.g., Miller's sensing method (Miller, 2006). This approach cannot provide a relationship between the incident field and source amplitudes however, so this was addressed in (Vasquez , 2009a,b, 2011), using multipole sources to create silent zones in two dimensions (2D). Further progress came in (Norris , 2012), where closed-form explicit formulas for the active source coefficients were deduced. This was also extended to elastodynamics (Futhazar , 2015; Norris , 2014). This approach is attractive because the active source coefficients are independent of the object to be cloaked. This is in contrast to object-dependent approaches, which must be modified in resonant regimes (O'Neill , 2016). The latter approach, however, does not suffer from large amplitudes in the vicinity of the active source regions.

Although there has been extensive work on active manipulation of sound in three dimensions (3D) (Ahrens, 2012; Egarguin , 2020; Elliott , 2012; Onofrei and Platt, 2018), the object-independent approach has thus far been conducted in 2 D only, with the sole exception of Vasquez (2013), which considered the Helmholtz equation in 3D, where results were provided for the case of four active sources.

Here, we provide a framework for active cloaking in 3D. We derive expressions for the active source coefficients and introduce a fast, efficient methodology by placing active sources on vertices of the Platonic solids and exploiting their symmetry. Regardless of the incident field, once active source coefficients have been deduced for *one* of the active sources, the remaining source amplitudes can subsequently be determined, thus providing a complete exposition of 3D active cloaking for the Helmholtz equation.

## 2. Methodology

We consider active exterior cloaking for time-harmonic waves (with dependence $ e \u2212 i \omega t$, where *ω* is the angular frequency and *t* is time) governed by the 3D, homogeneous Helmholtz equation $ ( \u2207 2 + k 2 ) u ( x ) = 0$, where $ k = \omega / c$ is the wavenumber, with *c* the wavespeed. For acoustics, the scalar field $ u ( x )$ is the velocity potential at $ x = ( x , y , z )$.

*L*multipole active sources. The field subsequently scattered from an object interior to the active field is denoted by $ u s ( x )$. These fields are

*m*and degree

*n*in terms of the polar angle

*θ*and the azimuthal angle $\phi $ and

*θ*and $ \phi i$ are the incident wave angles. For plane wave incidence, the expansion coefficients $ Q n m = 4 \pi i n Y n m ( k \u0302 ) \xaf$ thus depend only on these incident angles, with overline denoting complex conjugation. For other types of incident fields, the values of

_{i}*Q*would be different, but the rest of the methodology presented below would be unchanged. The amplitude of the radiation mode with order (

_{nm}*n*,

*m*) in the $\u2113$ th active source (which is located at $ x \u2113$) is $ q \u2113 , n m$, where $ \u2113 = 1 , 2 , \u2026 , L$ and

*a*is the scattering coefficient of the object.

_{nm}To achieve active cloaking, $ q \u2113 , n m$ are sought such that for some closed domain *C* surrounded by the active sources with $ x \u2113 \u2209 C$, we have $ u i ( x ) + u d ( x ) = 0$ for $ x \u2208 C$ and $ u d ( x ) \u2192 0$ as $ | x | \u2192 \u221e$. Thus, while the scattered field from any object is nullified in *C*, we also stipulate that the radiation of *u _{d}* itself to the far field is zero, leaving minimal evidence of the cloak.

The active sources are located at the vertices of an imaginary Platonic solid, thus limiting the number of sources *L* to five values (Euclid, 2012) [see Figs. 2(a)–2(e)]. This geometry ensures that the sources reside on the circumsphere of the Platonic solids such that $ | x \u2113 | = x 0$ for all $\u2113$ in each case, where *x*_{0} is an arbitrary constant. We set $ x 1 = ( 0 , 0 , \u2212 x 0 )$ in every case such that the active source with $ \u2113 = 1$ always has the lowest *z*-coordinate. To locate the remaining sources, we note that a Platonic solid consisting of *q p*-sided regular polygonal faces around each vertex can be characterized by two indices (*p*, *q*) [e.g., a cube with three squares around each vertex has the indices (4, 3)]. We define the length of each side of a Platonic solid as *sx*_{0}.

*p*,

*q*) and

*s*, whose values are listed in Table 1 (see Part 2 of the supplementary material

^{1}). Since the source distribution is

*q*-fold rotationally symmetric about the

*z*-axis, if we assign the indices $ \u2113 = 2 , \u2026 , q + 1$ to the

*q*vertices located immediately above the source $ \u2113 = 1$ in the counterclockwise direction, their position vectors $ x \u2113$ are given by

L
. | (p, q)
. | s
. | $v$ (in units of $ x 0 3$) . |
---|---|---|---|

4 | (3, 3) | $ 2 6 / 3$ | 0.0033 |

6 | (3, 4) | $ 2$ | 0.0675 |

8 | (4, 3) | $ 2 3 / 3$ | 0.0538 |

12 | (3, 5) | $ 2 5 ( 3 \u2212 \varphi ) / 5$ | 0.4562 |

20 | (5, 3) | $ 2 3 ( \varphi \u2212 1 ) / 3$ | 0.3692 |

L
. | (p, q)
. | s
. | $v$ (in units of $ x 0 3$) . |
---|---|---|---|

4 | (3, 3) | $ 2 6 / 3$ | 0.0033 |

6 | (3, 4) | $ 2$ | 0.0675 |

8 | (4, 3) | $ 2 3 / 3$ | 0.0538 |

12 | (3, 5) | $ 2 5 ( 3 \u2212 \varphi ) / 5$ | 0.4562 |

20 | (5, 3) | $ 2 3 ( \varphi \u2212 1 ) / 3$ | 0.3692 |

*C*, we employ the procedure from Vasquez (2013) as the starting point and use the approach in Norris (2012) to obtain explicit expressions (see Part 1 of the supplementary material

^{1}) in the form

*Q*and is, therefore, valid for any angles of incidence. In Eq. (7), $ S \u0302 t \nu s \mu ( x \u2113 )$ [defined in Eqs. (8)–(11) of the supplementary material

_{ts}^{1}] is a coefficient depending on the position vector $ x \u2113$, and the quantities $ D \nu n , i n \nu m \mu ( x \u2113 , \u2202 C \u2113 )$ are

^{1}). The surface integral $ i n \nu m \mu $ is evaluated over the face $ \u2202 C \u2113$ and parameterized by the vector $ y \u2212 x \u2113$ for $ y \u2208 \u2202 C \u2113$. It appears that an explicit form of $ i n \nu m \mu $ is not available; however, the integral can be simplified such that its numerical evaluation becomes significantly less expensive (see Part 4 of the supplementary material

^{1}). The formulations in Eqs. (6)–(10) hold when the domain

*C*is completely bounded by the union of faces $ \u2202 C \u2113$, where $ \u2113 = 1 , 2 , \u2026 , L$, and $ | y \u2212 x \u2113 | = a x 0$ for $ y \u2208 \u2202 C \u2113$ such that $ \u2202 C \u2113$ is a surface belonging to a sphere centered at $ x \u2113$ with radius $ a x 0$. The cloaked region

*C*therefore consists of the domain interior to the union of surfaces formed by identical imaginary spheres of radius $ a x 0$ located at the vertices and the source amplitude in the form of Eqs. (6)–(10) applies to a general incident wave. The case of plane wave incidence [ $ Q n m = 4 \pi i n Y n m ( k \u0302 ) \xaf$] admits the following compact form:

^{1}we discuss how to choose this truncation parameter such that the source amplitude converges with a prescribed level of accuracy.

In Figs. 2(a)–2(e), we illustrate the respective cloaked regions inside each of the Platonic solids with *a* taken as the lower bound in Eq. (10) (when the volume of the domain is a maximum). In each case, the surface of integration $ \u2202 C \u2113$ is delimited by *q* identical circular arcs (*q* depends on *L* as indicated in Table 1). For non-Platonic distributions of sources, these arcs are *not* identical.

^{1}). Since $ \u2202 C 1$ possesses

*q*-fold rotational symmetry about the

*z*-axis, it can be subdivided into

*q*congruent segments, each being the region bounded by

*C*by using the divergence theorem on Eqs. (12)–(15).

*L*active sources. We write the coefficient $ q \u2113 , n m$ as $ q \u2113 , n m ( k \u0302 , x \u2113 , \u2202 C \u2113 )$ to indicate its dependence on the propagating vector $ k \u0302$ and the two geometric parameters. For brevity, we also denote the surface integral as $ i \u2113$ by suppressing indices and arguments in Eq. (9). Define a new coordinate system $ x \u2032 = ( x \u2032 , y \u2032 , z \u2032 ) = R ( v \u0302 , \Theta ) x$ where $ R ( v \u0302 , \Theta )$ is a rotation matrix, with $ v \u0302$ being the unit vector representing the rotation axis and Θ, the rotation angle. As illustrated in Fig. 3, we rotate the original frame

**x**such that in the rotated frame $ x \u2032$, the $\u2113$ th active source, takes the bottom-most location, replacing the source $ \u2113 = 1$, and the corresponding bounding surface $ \u2202 C \u2113$ has the same orientation as $ \u2202 C 1$ in the original frame

**x**. Under this transformation, the amplitude in Eq. (11) becomes

**R**ensures that they remain rotationally invariant across all $\u2113$. In particular, we have $ D \nu n \u2032 = D \nu n$ and $ i \u2032 \u2113 = i 1$. The source amplitude $ q \u2113 , n m ( k \u0302 , x \u2113 , \u2202 C \u2113 )$, is thus transformed to $ q 1 , n m ( k \u2032 \u0302 , x 1 , \u2202 C 1 )$ in the rotated frame. While the former is calculated by integrating over $ \u2202 C \u2113$ for all $\u2113$, the latter can be evaluated by simply integrating over $ \u2202 C 1$ parameterized by Eqs. (12)–(15) and replacing $ k \u0302$ by $ k \u0302 \u2032$. If we express the source coefficients as a vector $ q \u2113 , n = { q \u2113 , n m} m = \u2212 n n$, then it can be shown that the system of linear equations

^{1}] is the Wigner D-matrix (Wigner, 2012) with dimensions $ ( 2 n + 1 ) \xd7 ( 2 n + 1 )$ and $ \gamma , \beta , \alpha $ are the Euler angles (Varshalovich , 1988) of the matrix $ R ( v \u0302 , \Theta )$ such that

*y*and

*z*axes. By solving Eq. (17) for $ n = 0 , 1 , \u2026 , N$ with

*N*a positive integer, we can retrieve the full set of amplitudes $ q \u2113 , n m ( k \u0302 , x \u2113 , \u2202 C \u2113 )$ up to the

*N*th order multipole. The cloaking device is thus devised by repeating this procedure with the corresponding $ k \u0302 \u2032$ and Euler angles for each source. Employing active sources on the vertices of the Platonic solids is beneficial because once one set of source coefficients is determined, those for the others follow from simple post-processing operations. See Part 5 of the supplementary material

^{1}for more details, including the exact form of the rotation matrix

**R**.

## 3. Results

The method is illustrated in Fig. 4, with *L* = 20 active sources distributed at the vertices of a regular dodecahedron with source distance $ k x 0 = 5 \pi $, which was produced using a Mathematica code that is available to download from https://github.com/himyeung1025/3d_silent_zone_cloaking. We consider the problem in the context of acoustics, but the principles are similar for other scalar waves. In Figs. 4(a)–4(d), the incident plane wave propagates in the positive *x* direction with $ \theta i = \pi / 2 , \phi i = 0$; in Fig. 4(e)–4(h), it has angles of incidence $ \theta i = \pi / 2 , \phi i = \pi / 4$. In Figs. 4(a), 4(b), 4(e), and 4(f), the scattering object inside the cloaked region *C* is a sound-soft sphere with radius $ k A = 3 ( 1 \u2212 a ) k x 0 \u2248 0.8579 \pi $, where $ a = ( 2 6 / 3 ) / [ 2 \u2009 sin \u2009 ( \pi / 3 ) ]$, and *a* is chosen to be the minimum permissible radius of the imaginary spheres bounding *C* for *L* = 4, which means that the scattering sphere has a radius three times that of the inscribed sphere of *C* when *L* = 4. In Figs. 4(c), 4(d), 4(g), and 4(h), we show the case where there is no scattering object. In all subplots, the real part of the total wave field *u* on the cross-section *z* = 0 is shown. In Figs. 4(a), 4(c), 4(e), and 4(g), the cloaking devices are inactive and a prominent scattering pattern, including distorted wavefronts and a shadow region behind the sphere, is observed. In Figs. 4(b), 4(d), 4(f), and 4(h), the cloaking devices are activated and the multipole order of each active source is taken as *N* = 10. The series expansions for the source coefficients in Eq. (11) are truncated such that the active field produced by each source is within 1% relative error. The spheres now reside in quasi-silent regions. The straighter wavefronts and the absence of the shadow region in the figures indicate that the incident wave is scattered only slightly by the sphere inside the quiet zone and the sources radiate little into the exterior of the silent region, demonstrating the cloak's effectiveness. The wave field diverges within small neighbourhoods of the active point sources. In practice, these large fields are confined within the finite-sized sources. We have cropped the plot at $ Re ( u ) = 1.2$.

*W*in the far field after ( $ u d \u2260 0$) and before (

*u*= 0) cloak activation (see Part 6 of the supplementary material

_{d}^{1}). In Fig. 5, we plot the SWL in the case of scattering from a sound-soft sphere for the source distributions depicted in Fig. 2. The order

*N*, the sphere radius

*A*and the direction of the incident wave are as in Fig. 4.

For each *kA*, the truncation parameter is chosen to achieve a relative error of less than 1% in the active field generated by each source. We require $ \sigma < 1$ and thus $ 10 \u2009 log \u2009 \sigma < 0$. For all cases, the SWL remains negative for the range of *kA* studied here with increased efficacy at lower frequencies. As the source number *L* increases the cloaking effect generally improves, with a maximum reduction of approximately 70 decibels achieved when *L* = 20 at $ k A \u2248 0.1716 \pi $. Further parameter studies can be found in Part 6 of the supplementary material.^{1}

## 4. Conclusion

In conclusion, we have formulated an efficient, 3D, active exterior cloaking strategy for the scalar Helmholtz equation. The Platonic distribution of the active sources means that we only need to determine the source amplitudes at one location; those at other locations follow from post-processing, exploiting the symmetry and regularity of the Platonic solids.

## Acknowledgments

This work was supported by a University of Manchester President's scholarship for Yeung (2017-21) and by the Engineering and Physical Sciences Research Council (EP/L018039/1) for Parnell. The authors have no conflicts of interest to declare. The data in this paper was generated using Mathematica code that is available to download at https://github.com/himyeung1025/3d_silent_zone_cloaking.

^{1}

See supplementary material at https://doi.org/10.1121/10.0019906 for detailed mathematical derivations of the active noise control model.

## REFERENCES

*Analytic Methods of Sound Field Synthesis*

*The Thirteen Books of the Elements*

*Active Control of Vibration*

*107 Multiple Scattering: interaction of Time-Harmonic Waves with N Obstacles*

*Active Control of Sound*

*Quantum Theory of Angular Momentum*

*Acoustic Metamaterials*

*Group Theory: And Its Application to the Quantum Mechanics of Atomic Spectra*